Solutions of Fractional Differential Type Equations by Fixed Point Techniques for Multivalued Contractions

This paper involves extended b − metric versions of a fractional diﬀerential equation, a system of fractional diﬀerential equations and two-dimensional (2D) linear Fredholm integral equations. By various given hypotheses, exciting results are established in the setting of an extended b − metric space. Thereafter, by making consequent use of the ﬁxed point technique, short and simple proofs are obtained for solutions of a fractional diﬀerential equation, a system of fractional diﬀerential equations and a two-dimensional linear Fredholm integral equation.


Introduction and Preliminaries
In the last years, the fractional calculus branch [1,2] has attracted great interest.ere exist many kinds of proposed fractional operators, for instance, we have the well-known Caputo, Riemann-Liouville, Grunwald-Letnikov derivative etc.Among all the papers dealing with fractional derivatives, fractional differential equations as an important research field have attained great deal of attention from many researchers (see [3][4][5][6][7][8]).
ere are many applications of the fractional topic in complex analysis, such as, in the sense of conformable derivatives and integrals, interesting results for fractional formulations of complex-valued functions of a real variable have been successfully introduced, which in turn open the door to the researchers to construct the theory of conformable integration by studying functions of a complex variable [9].On the other hand, the standard definition for the Atangana-Baleanu fractional derivative involves an integral transform with a Mittag-Leffler function, where the kernel can be rewritten as a complex contour integral, which can be used to provide an analytic continuation of the definition to complex orders of differentiation [10].ese lines are very important due to their applications in the field of natural science or engineering.
In the last few decades and in the branch of fractional differential equations, Riemann-Liouville and Caputo derivative ones are the mostly used.Note that several fractional differential equations have been resolved by using fixed point techniques. is paper is concerned with this fact when considering the class of extended b-metric spaces.
Let (R, ϖ) be a metric space.Denote by CB(R) a set of nonempty closed bounded subsets of R. Define the function where ϖ(ξ, en, ¥ is called the Hausdorff-Pompeiu metric.Consider e following can be deduced from the definition of δ.For all [ 1 , [ 2 , [ 3 ∈ CB(R), we have the following: In 2017, the concept of extended b-metric spaces has been initiated by Kamran et al. [11], by considering a control function at the right-hand side of the triangular inequality.
Definition 1 (see [11]).Let R be a nonempty set and θ: R × R ⟶ [1, ∞) be a given function.An extended b-metric is a function ϖ θ : R × R ⟶ [0, ∞) such that, for all η, ξ, σ ∈ R, we have the following: is (generalized) metric space has attracted many researchers where many real applications have been resolved.For more details, see [12][13][14][15].Some of the related topological concepts are as follows.
Definition 2 Let (R, ϖ θ ) be an extended b−metric space.Let ξ m   m≥0 be a sequence in R. where Example 1 (see [16]).Take R � and Definition 3 (see [17]).Denote by Ξ the set of functions Υ: R + ⟶ (0, 1] such that Clearly, Υ ∈ Ξ. e manuscript is organized as follows.In Section 2, some fixed point results in the class of extended b-metric spaces have been provided.We also present some useful examples.By using fixed point techniques, we solve in Section 3 a fractional nonlinear differential equation, we ensure in Section 4 the existence of a unique solution of a system of nonlinear fractional differential equations, and in Section 5, we establish that a two-dimensional linear Fredholm integral equation has a unique solution.At the end, in Section 6, we give a conclusion.

Main Theorems
In this section, R refers to an extended b−metric space equipped with the distance ϖ θ .We begin with the following lemmas.
e assertions (i)-(v) follow immediately by Czerwik [18] in b−metric spaces and (vi)-(vii) follow immediately by the definition of an extended b− metric space and (1) with (2).
Proof.By a similar way as in the proof of Lemma 4 in [19], we get the result.Now, we state and prove our main theorems.
Using Lemma 2, we get At the limit, we have ϖ θ (φ, Iφ) ⟶ 0. us, φ ∈ Iφ.Similarly, we can show that φ ∈ ℘φ.Hence, φ is a cfp of the two mappings ℘ and I.For the uniqueness, let ] ≠ φ be another cfp of ℘ and I, then, by our contractive condition, one can write i.e., the uniqueness holds.
en, the proof is completed.

Complexity
If we consider ℘ � I in the above theorem, we get the important below result.
If we put ℘ � I in eorem 2, we have the following result.
Hence, the conditions managed by eorem 2 are fulfilled, thereby concluding 0 ∈ R is the unique cfp of ℘ and I.

Complexity
Hence, the conditions managed by eorem 2 are fulfilled, thereby concluding 0 ∈ R is the unique cfp of ℘ and I.

Solving a Fractional Nonlinear Differential Equation
Recently, by the technique of nonlinear analysis such as fixed-point results, the Leray-Schauder theorem and stability, there are some papers dealing with the existence of solutions of nonlinear initial-value problems of fractional differential equations (see [21][22][23]). e main advantage of using fractional nonlinear differential equations is to describe the dynamics of complex nonlocal systems with memory.is part is devoted to obtain an existence solution of the subsequent nonlinear differential equation of fractional order: with boundary conditions e Caputo fractional derivative ∁ D ϑ with ordered ϑ is defined as follows: for all ξ, ℓ ∈ R, e > 1. en, the pair (R, ϖ θ ) is a complete extended b−metric space [20].Here, we need to be reminded that the Riemann-Liouville fractional integral of order ϑ is as follows: Now, our main theorem of this section is.

Theorem 3.
e problem (36) with boundary conditions (37) has a unique solution if the following assumptions are fulfilled: (ii) ere is a constant ϕ such that ϕG < 1, where Proof.Define the mapping ℘: R ⟶ R by , ξ is a unique fixed point of the multivalued mapping ℘.To get that, we shall prove that ℘ satisfies the contractive condition of Corollary 1.Consider 8 Complexity is implies that where κ � 2 e− 1 , Ω(ϰ) � 4 e− 1 ϰ, and Λ(ϰ) � (ϰ/16 e− 1 ).en, by Corollary 1, there exists a unique fixed point of the mapping (43), which is the unique solution of problem (36) in R.

Solving a System of Nonlinear Fractional Differential Equations
Differential equations of fractional order have been the focus of many studies due to their frequent appearance in various applications in fluid mechanics, viscoelasticity, physics, engineering, and biology.Recently, a large amount of literature studies developed concerning the application of fractional differential equations in nonlinear dynamics [24][25][26][27][28][29].In this part, we shall find the existence of a solution to the following system of nonlinear fractional ordered differential equations: where Ω and Ω * are constants, ℘, I: [0, 1] × R + ⟶ R + , and ∁ D ϑ refers to the Caputo fractional derivative.If we apply Green's (continuous) function Υ(τ, μ) on [0, 1] × [0, 1], ( 18) is equivalent to the following system: where Υ(τ, μ) is defined by and ϕ 2 (τ, μ, ℓ(μ)) � Υ(τ, μ)I(ℶ(μ)), then system (19) turns into for all ξ, ℓ ∈ R, e > 1. en, the pair (R, ϖ θ ) is a complete extended b−metric space.Now, we provide the following theorem to derive an existence result for the solution of problem (49).Theorem 4. Consider system (49) under the following hypotheses: en, problem (49) has a unique solution on R, which is considered as the unique solution to system (46).Proof.Consider two multivalued mappings ℘, I: R ⟶ R having the form Complexity Hence, the unique cfp of the mappings ℘ and I defined in (52) is considered as the unique solution of problem (49), which leads to the solution of problem (47) and from it to the solution of system (46).Consider where κ � 2 e− 1 , Ω(ϰ) � 4 e− 1 ϰ, and Λ(ϰ) � (ϰ/16 e− 1 ).us, by eorem 1, there exists a unique cfp of the mappings (52), which is the unique solution of system (46) in R. □

Solving a Two-Dimensional Linear Fredholm Integral Equation
Two-dimensional Fredholm integral equations of the second kind arise from many problems in engineering and mechanics by a suitable transformation.For example, in the calculation of plasma physics, it is usually required to solve Fredholm integral equations (see [30]).

Conclusion
In this manuscript, we gave some common fixed point theorems involving generalized multivalued contraction mappings in the class of extended b-metric spaces.Applying our obtained results, we ensure the existence of (unique) solutions of a fractional differential equation, a system of fractional differential equations and a two-dimensional linear Fredholm integral equation.is affirms the utility of fixed point theory in the framework of fractional calculus.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.